12 problems

Let $Q(x)=x^T A x$ be a positive definite quadratic form with $A$ symmetric, let $\Delta_Q=Q(\partial_1,\partial_2,\partial_3)$ on $\mathbb{T}^3=\mathbb{R}^3/\mathbb{Z}^3$, and def…

Unstable periodic geodesics conjecture. The particular geometry of unstable periodic geodesics should play a key role in spectral-projector estimates, especially in determining opt…

Chabert's spectral-projector conjecture. The identity above should remain valid up to $\delta=\lambda^{-\frac13}$.

Quantum complete integrability conjecture. There exists $\kappa=\kappa(N,d)>0$ such that

Sharp bandwidth conjecture. The main projector estimate should hold for

Smoothed-projector expansion conjecture. Roughly, one should have

Branching-point contribution conjecture. The contribution should satisfy

Germain's conjecture. One should have

Interpolation restriction conjecture. Nontrivial improvements over the case $\delta=1$ should be possible only for $p=\infty$, or for cases obtained from $p=\infty$ by interpolatio…

Let $M$ be the Riemannian manifold under consideration, let $P_{\lambda,\delta}$ denote the spectral projector associated with frequencies in a window of size $\delta$ around…

Gutiérrez–Maldonado's conjecture. For an arbitrary torus,

On a Zoll manifold, the square roots of the Laplace eigenvalues form clusters concentrating along an arithmetic progression, with the width of the $k^{th}$ cluster on the order of…